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ਕਦਮ ਦਰ ਕਦਮ

1. \int_{0}^{\infty} x e^{- x}\, dx

   First find an antiderivative F, then evaluate F(b) − F(a).

2. u = - x,\quad du = -1\, dx

   Substitute u = - x.

3. \int x e^{- x}\, dx = \int u e^{u}\, d_u

   Rewrite the integral in terms of u.

4. u = u,\quad dv = e^{u}\, d_u

   Integration by parts: ∫u dv = uv − ∫v du.

5. du = 1\, d_u,\quad v = e^{u}

   Differentiate u, integrate dv.

6. \int u e^{u}\, d_u = u e^{u} - \int e^{u}\, d_u

   Apply the formula.

7. \int e^{u}\, d_u = e^{u}

   ∫ aᵘ du = aᵘ / ln a (for eˣ that is just eˣ).

8. = u e^{u} - e^{u}

   Combine.

9. = - x e^{- x} - e^{- x}

   Substitute back u = - x.

10. F(\infty) - F(0) = \left(\text{NaN}\right) - \left(-1\right)

    Fundamental theorem of calculus: plug in the limits.

11. = 1

    Simplify.

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